Low rank differential equations for Hamiltonian matrix nearness problems

For a Hamiltonian matrix with purely imaginaryeigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis.Conversely, for a Hamiltonian matrix with all eigenvalues lying offthe imaginary axis, we look for a nearest Hamiltonian matrix that hasa pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearnessproblems are motivated by applications such as the analysis of passivecontrol systems. Theyare closely related to the problem of determining extremalpoints of Hamiltonian pseudospectra. We obtain a characterization ofoptimal perturbations, which turn out to be of low rank and areattractive stationary points of low-rank differential equations thatwe derive. We use a two-level approach, where in the inner level we determine extremal points of the Hamiltonian $eps$-pseudospectrum for a given $eps$ by following the low-rank differential equations into a stationary point, and on the outer level we optimize for~$eps$. This permits us to give fast algorithms - exhibiting quadratic convergence - for solving the considered Hamiltonian matrixnearness problems.